

"An artist
is a person who knows how to make puzzles out of solutions."
Karl Kraus 



Christina Brech, Piotr Koszmider; l_{∞}sums and the Banach space l_{∞}/c_{0};
Fund. Math. 224 (2014), 175185 










We prove that the use of the Continuum Hypothesis in some results of Drewnowski and Roberts
concerning the Banach space l _{∞}/c _{0} cannot be avoided.
In particular, we prove that in the Cohen model, l _{∞}(c _{0}(c)) does not embed isomorphically into
l _{∞}/c _{0} where c
is the cardinality of the continuum. It follows that consistently l _{∞}/c _{0}
is not isomorphically of the form l _{∞}(X) for any Banach space X.
 














H. G. Dales, T. Kania, T. Kochanek, P. Koszmider, N. J. Laustsen,
Maximal left ideals of the Banach algebra of bounded operators on a Banach space; Studia Math. 218 (2013), 245286 










We address the following two questions regarding the maximal left ideals of the Banach algebra
B(E) of bounded operators acting on an infinitedimensional Banach space E:
 Does B(E) always contain a maximal left ideal which is not finitely generated?
 Is every finitelygenerated, maximal left ideal of B(E) necessarily of the form
{T in B(E) : Tx = 0} (*) for some nonzero x E?
Since the twosided ideal F(E) of finiterank operators is
not contained in any of the maximal left ideals given by (*),
a positive answer to the second question would imply a positive answer to the first.
Our main results are:
 Question (I) has a positive
answer for most (possibly all) infinitedimensional Banach spaces;
 Question (II) has a positive answer if and only if no finitelygenerated,
maximal left ideal of B(E) contains F(E);
 the answer to Question (II) is positive for many, but not all, Banach spaces.
 














Antonio Avilés, Piotr Koszmider;
A continuous image of a RadonNikodym compact space which is not RadonNikodym;
Duke Math. J. 162, 12 (2013), 22852299. 










We construct a continuous image of a RadonNikodym compact space which is not RadonNikodym compact, solving the problem posed in the 80ties by Isaac Namioka.
 














Antonio Avilés, Piotr Koszmider; A Banach space in which every injective operator is surjective;
Bull. London Math. Soc. (2013) 45 (5): 10651074 










We construct an infinite dimensional Banach space of continuous functions C(K) such that every onetoone operator on C(K) is onto.
 














Piotr Koszmider, Saharon Shelah;
Independent families in Boolean algebras with some separation properties;
Algebra Universalis 69 (2013), no. 4, 305  312 










We prove that any Boolean algebra with the subsequential completeness
property contains an independent family of size continuum. This improves a result of
Argyros from the 80ties which asserted the existence of an uncountable independent family.
In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties.
It follows that the Stone spaces of all such Boolean algebras contains a copy of the CechStone
compactification of the integers and the Banach space of contnuous functions on them has linfinity as a quotient.
Connections with the Grothendieck property in Banach spaces are discussed.
 














Piotr Koszmider; On large indecomposable Banach spaces; J. Funct. Anal. 264 (2013), no. 8, 1779–1805 










Hereditarily indecomposable Banach spaces may have density at most
continuum (PlichkoYost, ArgyrosTolias). In this paper we show that this cannot be proved for
indecomposable Banach spaces. We provide the first example of an indecomposable
Banach space of density two to continuum. The space exists consistently,
is of the form C(K) and
it has few operators in the sense that any bounded linear operator T on C(K)
satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly
compact (strictly singular).
 














Jesus Ferrer, Piotr Koszmider, Wieslaw Kubis; Almost
disjoint families of countable sets and separable complementation properties; J. Math. Anal. Appl. 401 (2013), no. 2, 939–949 










We study the separable complementation property (SCP) and its natural
variations in Banach spaces of continuous functions over compacta K_{A} induced by almost
disjoint families A of countable subsets of uncountable sets. For these spaces,
we prove among others that C(K_{A}) has the controlled variant of the separable
complementation property if and only if C(K_{A}) is Lindelof in the weak topology
if and only if K_{A} is monolithic. We give an example of
A for which C(K_{A}) has the SCP, while K_{A}
is not monolithic and an example of a space C(K_{A}) with controlled
and continuous SCP which has neither a projectional skeleton nor a projectional
resolution of the identity. Finally, we describe the structure of almost
disjoint families of cardinality omegaone which induce monolithic spaces of
the form K_{A}: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.
 














Christina Brech, Piotr Koszmider;
On universal spaces for the class of Banach spaces
whose dual balls are uniform Eberlein compacts; Proc. Amer. Math. Soc. 141 (2013), 12671280 










For k being the first uncountable cardinal w_{1} or
k being the cardinality of the continuum c, we prove that
it is consistent that there is no Banach space of density k
in which it is possible to isomorphically embed every Banach space
of the same density which has a uniformly Gateaux differentiable renorming
or, equivalently, whose dual unit ball with the weak^{*} topology is a subspace of
a Hilbert space (a uniform Eberlein compact space). This complements a consequence of
results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that
assuming the continuum hypothesis, there is a universal space for all Banach spaces
of density k=c=w_{1}
which have a uniformly Gateaux differentiable renorming.
Our result implies, in particular, that betaNN
may not map continuously onto a compact subset of a Hilbert space
with the weak topology of density k=w_{1} or
k=c and that a C(K) space for some
uniform Eberlein compact space K may not embed isomorphically into l_infty/c__{0}.
 














Piotr Koszmider;
A C(K) Banach space which does not have the SchroederBernstein property; Studia Math. 212 (2012), 95117 










We construct a totally disconnected compact Hausdorff space
N which has clopen subsets M included in L included in N such that
N is homeomorphic to M and hence C(N) is isometric as a Banach space
to C(M) but
C(N) is not isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form
C(K) which are isomorphic to complemented subspaces of each other (even in the above strong
isometric sense),
providing a solution to the SchroederBernstein problem for Banach spaces
of the form C(K). N is obtained as
a particular compactification of the pairwise disjoint union of a sequence of Ks for which
C(K)s have few operators.
 














Christina Brech, Piotr Koszmider;
On universal Banach spaces of density continuum, Israel J. Math. 190 (2012), 93–110. 










We consider the question whether there exists a Banach space X of density continuum such that
every Banach space of density not bigger than continuum
isomorphically embeds into X (called a universal
Banach space of density continuum).
It is well known that linfinity by czero is such a space if we assume the continuum hypothesis.
However, some additional settheoretic assumption is needed, as we prove in the main result of
this paper that
it is consistent with the usual axioms of settheory
that there is no universal Banach space of density continuum.
Thus, the problem of the existence of a universal Banach space of density continuum
is undecidable using the usual axioms of settheory.
We also prove that it is consistent that there are universal
Banach spaces of density continuum, but linfinity by czero is not among them.
This relies on the proof of the consistency of the nonexistence
of an isomorphic embedding of C([0,c]) into
linfinity by czero.
 














Piotr Koszmider; Some topological invariants and biorthogonal systems in Banach spaces,
Extracta Math. 26(2) (2011), 271294 










We consider topological invariants on compact spaces
related to the sizes of discrete subspaces (spread), densities of subspaces,
Lindelof degree of subspaces, irredundant families of clopen sets and others
and look at the following associations between compact topological spaces and Banach spaces:
a compact K induces a Banach space C(K) of real valued
continuous functions on K with the supremum norm; a Banach
space X induces a compact space B _{X*}, the dual ball with
the weak ^{*} topology. We inquire on how topological invariants on
K and B _{X*} are linked to the sizes of biorthogonal systems and
their versions in C(K) and X respectively. We gather folkloric facts and
survey recent results like that of LopezAbad and Todorcevic that it is
consistent that there is a Banach space X without uncountable
biorthogonal systems such that the spread of
B _{X*} is uncountable or that of Brech and Koszmider that
it is consistent that there is a compact space where spread of K ^{2}
ic countable but C(K) has uncountable biorthogonal systems.
 














Christina Brech, Piotr Koszmider;
On biorthogonal systems whose functionals are finitely supported;
Fund. Math. 213 (2011), 4366 










We show that for each natural n>1 it is consistent that
there is a compact Hausdorff space K_{2n}
such that in C(K_{2n}) there is no uncountable (semi)biorthogonal
sequence (f_{i}, m_{i}) for i in w_{1}where m_{i}'s are atomic measures
with supports consisting of at most 2n1 points of K_{2n}, but there are
biorthogonal systems (f_{i}, m_{i}) for i in w_{1}where m_{i}'s are atomic measures
with supports consisting of 2n points.
This complements a result of Todorcevic that Martin's axiom with the negation of CH implies that each
nonseparable Banach space C(K) has an uncountable biorthogonal system where
the functionals are differences of two pointwise measures.
It also follows that
it is consistent that the irredundance of
the Boolean algebra Clop(K) or the Banach algebra C(K)
for K totally disconnected can be strictly smaller
than the sizes of biorthogonal systems in C(K).
The compact spaces exhibit an interesting behaviour with respect to
known cardinal functions: the hereditary density of the powers K_{{2n}}^{k}
is countable up to k=n and it is uncountable (even the spread
is uncountable) for k>n.
 














Piotr Koszmider, Miguel Martín, Javier Merí;
Isometries on extremely noncomplex Banach spaces;
Journal of the Institute of Mathematics of Jussieu, 10 (2011) No. 02, pp.325348 










We construct an example of a real Banach space whose group of
surjective isometries reduces to plus or minus identity, but the group of
surjective isometries of its dual contains the group of
isometries of a separable infinitedimensional Hilbert space as
a subgroup. To do so, we present examples of extremely
noncomplex Banach spaces (i.e. spaces X such that Id+
T^{2}=1+T^{2} for every bounded linear operator T on X)
which are not of the form C(K), and we study the surjective
isometries on this class of Banach spaces.
 














Christina Brech, Piotr Koszmider;
Thinvery tall compact scattered spaces which are hereditarily separable;
Transactions of the American Mathematical Society; 363 (2011), no. 1, pp. 501  519 










We strengthen the property Delta
of a function f from pairs of omega2 to countable subsets of omega2
considered by Baumgartner and Shelah. This allows us to consider new type of amalgamations
in the forcing used by Rabus, Juhasz and Soukup to construct thinvery tall compact scattered spaces.
We consistently obtain spaces K as above where nth power of K is hereditarily separable for each natural N.
This serves as a counterexample concerning cardinal functions on compact spaces as well
as has some applications in Banach spaces: the Banach space C(K) is an Asplund
space of density aleph2 which has no Frechet smooth renorming nor an
uncountable biorthogonal system.
 














Piotr Koszmider, Przemys³aw Zieliñski;
Complementation and Decompositions in some weakly Lindelof Banach spaces;
J. Math. Anal. Appl. 376 (2011) 329–341 










We consider the questions if a
Banach space of the form C(K) of a given class
(1) has a complemented copy
of c _{0}(G) for G uncountable or (2)
for every c _{0}(G) in X has a complemented c _{0}(E) for
an uncountable E in G or (3) has a decomposition
X=A+B where both A and B are nonseparable.
The results concern
a superclass of the class of nonmerizable Eberlein compacts, namely
Ks such that C(K) is Lindelof in the weak topology and we
restrict our attention to Ks scattered of countable height.
We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms
which are independent of ZFC plusminus CH.
If we assume the Pideal dichotomy,
for every c _{0}(G) in C(K) there is a complemented $c _{0}(E) for
an uncountable E in G,
which yields the positive answer to the remaining questions.
If we assume the club axiom, then we construct a nonseparable
weakly Lindelof C(K) for K of height w+1
where every operator is of the form cI+S
for c real and S an operator with separable range and conclude from this that
there are no decompositions as above which yields the negative answer to all the above
questions.
Since, in the case of a scattered compact K, the weak topology on C(K) and the
pointwise convergence topology coincide on bounded sets, and so
the Lindelof properties of these two topologies are equivalent,
many results concern also the space C _{p}(K).
 














Piotr Koszmider;
A survey on Banach spaces C(K) with few operators; RACSAM 104 (2), 2010, pp. 309 326 










We say that a Banach space C(K) has few operators if
for every operator T on C(K) we have T=gI+S or
T^{*}=g^{*}I+S where g is continuous on K, g^{*} is Borel on K
and S are weakly compact on C(K) or C^{*}(K) respectively.
Banach spaces of continuous functions with few operators
provided solutions to several long standing open problems
in the theory of Banach spaces.
The class of spaces is being gradually illuminated and applied further in the recent work
of P. BorodulinNadzieja, R. Fajardo, V. Ferenczi,
E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow and the author.
We describe basic properties, applications and relevant open problems.
 














Piotr Koszmider;
On a problem of Rolewicz about
Banach spaces that admit support sets;
Journal of Functional Analysis 257 (2009) pp. 27232741 










We construct
an example of a nonseparable Banach space
which does not admit a support set.
It is a consistent (and necessarily independent from the axioms of ZFC) example
of a
space $C(K)$ of continuous functions
on a compact Hausdorff $K$ with the
supremum norm. The construction depends on a construction
of a Boolean algebra with some combinatorial properties.
The space
is also hereditarily Lindelof in the weak
topology but it doesn't have
any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.
 














Istvan Juhasz, Piotr Koszmider, Lajos Soukup;
A first countable, initially w_{1}compact, but noncompact space;
Topology and its Applications 156 (2009) pp. 18631879 










We force a first countable, normal, locally compact,
initially w_{1}compact but noncompact space
X of size w_{2}.
The onepoint compactification of X is a nonfirst countable
compactum without any (nontrivial) converging w_{1}sequence.
 














Artur Bartoszewicz and Piotr Koszmider;
When an atomic and complete algebra of sets is a field of sets with nowhere dense
boundary; Journal of Applied Analysis 15 (2009) pp. 119127 










We consider pairs
where A is an algebra of sets from some class
called the class of algebras of type and where H(A)
is the ideal of hereditary sets of A. We characterize which of the above pairs
are topological, that is, which
are fields of sets with nowhere dense boundary for some topology together with
the ideal of nowhere dense sets for this topology.
Making use of the BalcarFranek theorem we construct an example of a pair
with complete quotient algebra and the hull property but not
topological. This countrexample, given in ZFC, provides the complete solution
of a problem posed in [M. Balcerzak, A.Bartoszewicz, K.Ciesielski,
Algebras with inner MBrepresentation, 29(1) 20032004, Real. Anal. Exchange.]
Such an algebra was constructed in [A. Bartoszewicz,
On some algebra of sets in Steprans strongQsequence model, Topology Appl.
149, (2005), no. 13, 915.]
under some aditional set theoretic assumption.
 














Piotr Koszmider, Miguel Martín, Javier Merí;
Extremely noncomplex C(K) spaces;
Journal of Mathematical Analysis and Applications
Volume 350, Issue 2, 15, 2009, pp. 601615 










We show that there exist infinitedimensional extremely
noncomplex Banach spaces, i.e. spaces X such that the norm
equality Id + T^{2}=1 + T^{2} holds for every bounded
linear operator T from X to X. This answers in the
positive Question 4.11 of [Kadets, Martín, Merí; Norm
equalities for operators, Indiana U. Math.
J. 56 (2007), 23852411]. More concretely, we show
that this is the case of some C(K) spaces with few operators
constructed in [Koszmider, Banach spaces of continuous
functions with few operators, Math. Ann. 330
(2004), 151183] and [Plebanek, A construction of a Banach
space C(K) with few operators, Topology
Appl. 143 (2004), 217239]. We also construct
compact spaces K _{1} and K _{2} such that C(K _{1}) and C(K _{2})
are extremely noncomplex, C(K _{1}) contains a complemented
copy of C(2 ^{w}) and C(K _{2}) contains a (1complemented)
isometric copy of linfinity.
 














Piotr Koszmider;
The interplay between compact spaces and the Banach spaces of their continuous functions;
in Open Problems in Topology 2; ed. Elliott Pearl, Elsevier 2007. 










Many open problems in the isomorphic Banach space theory are related to
the C(K)s and are left untouched sometimes for many decades. Some
infinitary combinatorial ideas developed in the last three decades, so
successful in the context of Ks, often have not yet been tested on the
C(K)s. What follows aims at suggesting the tests.
 














Piotr Koszmider;
Kurepa trees and topological nonreflection;
Topology and Its Applications, vol 151, 2005, No. 1., pp. 77  98. 










A property P of a structure S
does not reflect if no substructure of S of smaller cardinality
than S has the property.
If for a given property P
there is such an S of cardinality k,
we say that P does not reflect at k.
We undertake a fine analysis
of Kurepa trees which results in defining canonical topological and
combinatorial structures associated
with the tree which possess a remarkably wide range of
nonreflecting properties providing new constructions and solutions
of open problems in topology.
The most interesting results show
that many known properties may not reflect
at any fixed singular cardinal of uncountable
cofinality. The topological
properties we consider vary from normality, collectionwise
Hausdorff property to metrizablity and many others. The
combinatorial properties are related to stationary reflection.
 














Piotr Koszmider;
Projections in weakly compactly
generated Banach spaces and Chang's Conjecture;
Journal of Applied Analysis, 11 (2005), No. 2, 187205
 









Classical results on weakly compactly generated (WCG)
Banach spaces imply the existence of projectional resolutions of
identity (PRI) and the existence of many
projections on separable subspaces (SCP). We address the
questions if these can be the only projections in a nonseparable WCG space,
in the sense that there is a PRI of projections P\alpha's
for alpha between omega and lambda
such that any projection
is the sum of an operator in the closure of the linear span of countably
many Palpha's (in the strong operator topology)
and a separable range operator.
Wark's modification of Shelah's and Steprans' construction provides
an unconditional example for lambda equal to omega_1. We note that
it is impossible for lambda biger than omega_2.
The main
result of the paper is that for lambda equal omega_2, the second uncountable cardinal,
the question is logically undecidable and depends on
additional axioms deciding
the combinatorics
on omega_2; for example
Chang's conjecture implies that
there are other projections than the projections
mentioned above. The full strength results concern
all linear operators not just the projections.
 














Piotr Koszmider;
A space C(K) where all nontrivial complemented
subspaces have big densities;
Studia Mathematica 168 (2005), pp. 109  127 










Using the method of
forcing we prove that consistently there is
a Banach space (of continuous functions on a
zerodimensional compact Hausdorff space)
of density k bigger than the continuum
where all operators are multiplications
by a continuous function plus a weakly compact operator
and which has no
infinite dimensional complemented
subspaces of density smaller or equal to the continuum.
In particular no separable infinite dimensional
subspace has a complemented superspace of density
smaller or equal to the continuum,
consistently answering a question of Johnson and Lindenstrauss of 1974.
 














Piotr Koszmider;
On Decompositions of Banach spaces of continuous
functions on Mrówka's spaces; Proceedings of the American Mathematical
Society, 133 (2005), pp. 2137  2146. 










It is wellknown that if
K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets)
then the Banach space C(K)
of continuous functions on K has
complemented copies of c_{0}.
We address the question if it could be the
only type of decompositions of C(K) not isomorfic to c_{0}
into infinite dimensional summands for K infinite, scattered.
Making a special settheoretic assumption like the
continuum hypothesis or Martin's axiom we construct an example
of Mrówka's space (i.e., obtained from an
almost disjoint family of sets of positive integers) which answers
positively the above question,
providing apparently first examples of Banach spaces not isomorphic to c_{0} whose
only nontrivial decompositions are into c_{0} and itself.
The proofs use characterizations of operators as scalar
multiples of the identity plus an operator
with the range included in a copy of c_{0}, i.e.,
our space has minimal possible space of
operators among C(K)'s different than c_{0} for scattered K.  














Piotr Koszmider;
Banach spaces of continuous functions
with few operators; Matchematische Annalen, vol 330, (2004)
No 1. pp 151  183 










We present two constructions of
infinite, separable, compact Hausdorff spaces K
for which the Banach space C(K)
of all
continuous realvalued functions with the supremum norm
has remarkable properties. In the first construction
K is zerodimensional and C(K)
is nonisomorphic to any of its proper subspaces
nor any of its proper quotients. In particular,
it is an example of a C(K) space where
the hyperplanes,
one codimensional subspaces of C(K), are not isomorphic to C(K).
In the second
construction
K is connected and C(K) is
indecomposable which implies that it is
not isomorphic to any C(K') for K's zerodimensional.
All these properties follow from the fact
that there are few operators on our C(K)'s.
If we assume the continuum hypothesis the spaces
have few operators in the sense that
every linear bounded operator T: from C(K) into C(K)
is of the form gI+S where g is in C(K) and S is weakly compact
or equivalently (in C(K) spaces) strictly
singular.
 














Marek Balcerzak, Artur Bartoszewicz, Piotr Koszmider;
On MarczewskiBurstin representable algebras;
Colloquium Mathematicum, vol 99, No 1, 2004 pp. 55  60 










We find
first unconditional
examples of algebras of sets which are not MBrepresentable. On the other hand,
we prove that every Boolean algebra is isomorphic to an MBrepresentable
algebra of sets.
 














Piotr Koszmider;
Universal matrices and strongly
unbounded functions;
Mathematical Research Letters;
Vol. 9, No 4. 2002, pp. 549  566 










Fix an uncountable cardinal l. A symmetric matrix
M=(m_{ab})_{a,b < l} whose entries are countable
ordinals is called
strongly universal if for
every positive integer n, for every
n×n matrix
(b_{i j})_{i,j < n} and for every uncountable set
A={a: a Î A} Í [l]^{n}
of disjoint ntuples a={a_{0},...a_{n1}}
there are a, a¢ Î A
such that
b_{ij}=m_{ai aj¢} for 0 < i, j < n.
We go beyond the recent dramatic discoveries
for l = w_{1}, w_{2}
and address the question of the
possibility of the existence of a strongly
universal matrix for l > w_{2}.
Due to the undecidibility of some weak versions of the Ramsey property for
l ³ w_{2} the positive answer can be at most
consistent, but we show that wellinvestigated methods
of forcing cannot yield that answer for l > w_{2}.
We use
our method of "forcing with side conditions in semimorasses"
to construct generically l by l
strongly universal matrices for any cardinal l.
The results are proved in
more generality,
related concepts are investigated, some questions are stated and
some application are given.  














Piotr Koszmider, Artur Tomita, Steven Watson;
Forcing countably compact group topologies on a larger free Abelian
groups; Topology Proceedings;
Vol. 25. 2002 pp. 563  574. 










D.Dikranjan and D.Shakmatov asked for
which cardinal k there exist a countably compact
group topology on the free Abelian group of size k.
M.Tkacenko obtained such group topology under the continuum hypothesis
for the free Abelian
group of size continuum. The second author
showed that under MA, such group topology could be obtained for the free Abelian
group of any size k equal to the continuum. Recently the second author
and S.Watson improved those
results to MAcountable. However it was still
open whether a free Abelian group of size bigger
than the continuum could be endowed with a
countably compact group topology.
The example below shows that this is true in a forcing model.
 














Piotr Koszmider, Franklin D.
Tall; A Lindelof space with no Lindelof
subspace of size alephone; Proceedings of the American Mathematical Society;
Vol. 130 (2002)
pp. 2777  2787. 










A consistent example of an uncountable Lindelöf
T_{3} (and hence normal) space with no
Lindelöf subspace of size alephone
is constructed.
It remains unsolved whether extra settheoretic assumptions
are necessary for the existence of such a space. However,
our space has size alephtwo
and is a Pspace, i.e., G_{d}'s
are open, and for such spaces
extra settheoretic assumptions turn out to be
necessary.  














Lúcia Junqueira, Piotr Koszmider;
On families of Lindelöf and related
subspaces of two to omega one.
Fundamenta Matematicae;
Vol. 169 (2001), no. 3, 205231. 










"We consider the families of all subspaces
of size omegaone of twotoomegaone
(or of a compact zero dimensional
space X of weight omegaone in general) which are normal, have
Lindelof property or are closed under limits
of convergent omegaonesequences. Various
relations among these families
modulo the club filter subsets of X are shown
to be consistently possible. One of the main tools is
dealing with a subspace of the form X_M for an elementary submodel
M of size omegaone. Various results with this flavor are obtained.
The other tool used is forcing and in this case various preservation
or nonpreservation
results of topological and combinatorial properties
are proved. In particular we prove that there
may be no c.c.c. forcing which destroys
Lindelof property of compact spaces, answering a question of Juhász.
Many related questions are formulated.  














Piotr Koszmider;
On Strong Chains of uncountable functions;
Israel Journal of Mathematics;
2000, Vol 118, 289  315. 










For functions f,g:w_{1}®w_{1},
where w_{1} is the first
uncountable cardinal, we write that
f << g if and only if {x Î w_{1}: f(x) is less or equal g(x)}
is finite. We prove the consistency of the existence
of a wellordered increasing << chain of length
w_{2}, solving a problem of A. Hajnal. The methods
involve previously developed by us forcing with
side conditions in morasses which
is a variation on Todorcevic's forcing with
models as side conditions. The paper is self contained
and requires from the reader the knowledge of Kunen's
textbook and some basic experience with proper forcing
and elementary submodels.  














Piotr Koszmider; Forcing minimal extensions of
Boolean algebras; 1999, Vol. 351, 30733117. 










We employ a forcing approach
to extending a Boolean algebras.
A link between some forcings and some cardinal functions
on Boolean algebras is found and exploited. We find the following
applications:
1) We make Fedorchuk's method more flexible, obtaining
for every cardinal l of
uncountable cofinality, a consistent example of
a Boolean algebra A_{l} whose every infinite homomorphic image
is of cardinality l and
has a countable dense subalgebra (i.e.,
its Stone space is a compact Sspace whose every infinite closed
subspace has weight l). In particular this construction
shows that it is consistent that the minimal character of
a nonprincipal ultrafilter in a homomorphic image
of an algebra A can be strictly less that the minimal
size of a homomorphic image of A,
answering a question
of J. D. Monk.
2) We prove that for
every cardinal of uncountable cofinality
it is consistent that 2^{w}=l and
both A_{l} and A_{w1} exist.
3) By combining these
algebras we obtain many examples that answer questions of
J.D. Monk.
4) We prove the consistency of MA + not CH + there is
a countably tight compact space without
a point of countable character, complementing results of
A. Dow, V. Malykhin, and I. Juhasz. Although the algebra
of clopen sets of the above space
has no ultrafilter which is countably generated, it is a subalgebra
of an algebra all of whose ultrafilters are countably generated.
This proves, answering a question of
Arhangel'skii, that it is consistent
that there is a first countable compact
space which has a continuous image
without a point of countable character.
5) We prove that for any cardinal l of
uncountable cofinality it is consistent that there is a
countably tight Boolean algebra
A with a distinguished ultrafilter ¥ such that
for every a\not ' ¥ the algebra Aa
is countable and
¥ has hereditary character l.  














Piotr Koszmider;
On the existenceof strong chains
in P(w_1)/Fin.
Journal of Symbolic Logic 1998, vol. 63. no. 3. 










(X_{a}:a in w_{2}) included in P(w_{1})
is a strong chain in P(w_{1})/Fin
if and only
if X_{b}X_{a} is finite and
X_{a}X_{b} is uncountable
for each b < a < w_{1}. We show
that it is consistent that a strong chain in
P(w_{1}) exists. On the other hand we show that
it is consistent that there is a strongly
almostdisjoint family in P(w_{1})
but no strong chain exists:
box_{w1}
is used to construct a c.c.c forcing that adds
a strong chain and Chang's Conjecture to prove that
there is no strong chain. We also show that
forcing a strong chain in w_{1}^{w1}/Fin would require
another method.  














Piotr Koszmider;
Applications of Rhofunctions;
In
Set Theory: Techniques and Applications
(eds. C. Di Prisco, J. Larson, J. Bagaria, A. Mathias) Kluwer
Acad. Publishers, Dotrecht, 1998, 8398. 










"We will provide below examples of these
canonical methods and by this we would like to defend the statement
that" rhofunctions are the most important functions
of twocardinal combinatorics"  














Piotr Koszmider; Models as side conditions;
In
Set Theory: Techniques and Applications
(eds. C. Di Prisco, J. Larson, J. Bagaria, A. Mathias)
Kluwer Acad. Publishers, Dotrecht,
1998, 99107. 










"The main idea
of the method of models as side conditions
is to construct a forcing notion whose conditions
p are of the form p=(D,N), where D is
the working part and comes from the original forcing
with finite conditions and N is an elementary chain
of countable elementary submodels."  














Gary Gruenhage, Piotr Koszmider;
Arhangel'skiiTall problem: A consistent
counterexample;
Fundamenta Matematicae;
1996, Vol 149, 275285. 










We construct a consistent examples of a normal locally compact
metacompact space which is not paracompact, answering
a question of A. V. Arhangelskii and F. Tall. An interplay
between a tower in P(w)/Fin, an almost disjoint
family in [w]^w, and a version of an (w,1)morass forms
the core of the proof. A part of the poset which forces
the couterexample can be considered a modification of
a poset due to Judah and Shelah
for obtaining a Qset by a countable support iteration.  














Gary Gruenhage, Piotr Koszmider;
Arhangel'skiiTall problem under Martin's Axiom;
Fundamenta Matematicae;
1995, Vol 149, 143166. 










We show that under MA(sigmacentred)(omegaone)
implies that normal locally compact metacompact spaces are paracompact,
and that MA(omega_1) implies normal locally compact
metalindelof spaces are paracompact. The latter result answers a
question of S. Watson. The first result implies that there is a model of
set theory in which all normal locally compact metacompact spaces are
paracompact, yet there is normal locally compact metalindelof
space which is not paracompact  














Katsuya Eda, Gary Gruenhage, Piotr Koszmider,
Kenichi Tamano, Stevo Todorcevic;
Sequential Fans in Topology; Topology and Its Applications,
1995, Vol 67, no 3, 189220. 










For an index set I, let S(I) be the sequential fan with I spines, i.e.,
the topological sum of I copies of the convergent sequence with
all nonisolated points identified.
The simplicity and the combinatorial nature of this space is what lies behind
its occurrences in many seemingly
unrelated topological problems.  














Piotr Koszmider;
Semimorasses and Nonreflection at Singular Cardinals; Annals of
Pure and Applied Logic;
1995, Vol 72, 123. 










Some subfamilies of P
_{k}(l), for
k regular, k <l, called (k,l)semimorasses are investigated. For l = k^{+},
they constitute weak versions of Velleman's simplified
(k,1)morasses,
and for l > k^{+}, they provide a combinatorial
framework which in some cases has similar applications to the application
of (k,1)morasses with this difference that the obtained
objects are of size > k^{+}, and not only
of size k^{+} as in the case of morasses.
New consistency results involve
(compatible with CH) existence of
nonreflecting objects of singular sizes of
uncountable cofinality such as a nonreflecting stationary set
in P
_{k}(l), a nonreflecting nonmetrizable space
of size l, a nonreflecting nonspecial tree of size
l. We also characterize
possible minimal sizes of nonspecial trees without
uncountable branches.
 














Piotr Koszmider; On Coherent
Families of FinitetoOne Functions;
Journal of Symbolic Logic, 1993, Vol. 58 (1), 128138. 










We consider the existence of
coherent families of
finitetoone functions on countable subsets of an
uncountable cardinal kappa.
The existence of such families for kappa
implies the existence of a winning 2tactic for player TWO
in the countablefinite game on kappa
answering a question of M. Scheepers. We prove that
coherent families exist on kappa=omega_n, where n\in\omega,
and that they consistently exist for every cardinal kappa.
We also prove that iterations of Axiom A forcings with
countable supports are Axiom A.
 














Piotr Koszmider;
On The Complete Invariance Property in Some Uncountable Products;
Canadian Mathematical Bulletin ; 1992, Vol. 35 (2),
221229. 










We consider uncountable products
of nontrivial compact, convex subsets of
Banach spaces. We show that these products
do not have the complete invariance property i.e. they include
a nonempty, closed subset which is not a fixed point set
(i.e. the set of all fixed points) for any continuous mapping
from the product into itself.
In particular we give an answer to W.Weiss' question
whether uncountable powers of the unit interval have the complete
invariance property.  














Piotr Koszmider;
A Formalism for Some Class of Forcing Notions;
Zeitschrift fur Mathematische Logic und Grundlagen der
Mathematik, 1992, Vol. 38, 413421. 










We introduce a class of forcing
notions of type S, which
contains among others, Sacks forcing, PrikrySilver forcing and their
iterations and products with countable supports. We construct
and investigate some formalism suitable for these forcing
notions, which allows all standard trics for iterations
or products with countable supports of Sacks forcing.
On the other hand, it does not involve internal combinatorial
structure of conditions of iterations or products. We prove
that the class of forcing notions of type S is closed
under products and certain iterations with countable supports.  














Winfried Just, Piotr Koszmider; Remarks
on Cofinalities and Homomorphism Types of Boolean Algebras;
Algebra Universalis 1991, Vol. 28, 138149. 










We show that it is consistent that
there are Boolean algebras of cardinalities smaller than the continuum which
have no infinite countable homomorphic images, which by the Stone
duality means that there can are compact Hausdorff spaces
with no convergent sequences of weights less than the continuum
(unlike the classical compact spaec with no converging sequences
the beta N). We also show that there are consistent examples
of Boolean algebras of
cardinalities smaller than the continuum whose cofinality is uncountable,
showing that the assumption of Martin's axiom is necessary
in a known result of S. Koppeleberg.
 














Piotr Koszmider;
The Consistency of the negation of CH and the
pseudoaltitude less or equal to omega one;
Algebra Universalis 1990, Vol. 27, 8087. 










We show that it is consistent that
the negation of the continuum hypothesis holds and all Boolean
algebras have pseudoaltitude less or equal to omegaone.
It follows that the following dichotomy is consistent:
Either a Boolean algebra has an uncountable independent family
or it has an infinite homomorphic image with an ultrafilter
of character less or equal to omega_1.
Another topological version
of a conlusion is: Either a compact separable Hausdorff space
maps onto a nonmetrizable Cantor cube or
has a nonisolated point of small character (omega or omega_1)
in the presence of arbitrarily large continuum.
 











